OK, thanks all for your assistance. Turns out the deck was pretty much all out of whack. I had to take it in to someone with a bit more of a clue than me, and they got it working right for the most part.
Apparently the blades weren't level with each other, and one side of the deck was higher than the other. They got all that fixed and the cut improved dramatically. One of the guides was contacting the belt, which was what was causing it to keep breaking. I ran it for 2+ hours yesterday with few issues, so I think that problem is fixed.

The only thing it's doing now is, every time I turn off the blades, the PTO belt drops off of the engine pulley. It's not a huge deal to turn the engine off, get under it and put it back on, but it is definitely annoying. I'm guessing I either need to tighten something up, or they put on a belt that is slightly too long.

I also learned a lesson about the risk of buying used - apparently this mower does not have the original engine on it. It's still a Kohler, but it's not the 27 hp EFI - it's closer to a 22 hp. It's still a good engine and the mower overall is in pretty good shape, so I don't plan on doing anything about it. It just means I have a bit less power, so I probably can't cut as fast as I should have been able to, but for what I need it for, it does just fine. Live and learn.

Now, I start it up and it runs great. I engage the blades and they come on like a jet engine. It cuts fine for about a lap, and then the blade speed starts dropping off dramatically. It gets to the point where it's spinning but barely cutting anything. I turn the mower off and wait a few minutes, then start it back up and the blade speed is back to normal - until I start cutting, when it does the same thing and slows down after a short while.
I DO hear some squeaking from the middle spindle, which makes me think one of the belts is slipping. I have spares, so I replaced both belts just to eliminate that as a cause, and it didn't make a difference. I tightened the spring on the handle that engages the blades, but I don't see what else I can adjust that might improve the situation.
If it means anything, the engine belt is hot when I touch it, the deck belt is not.
I looked under the deck and there is no grass buildup. The blades slow down even when cutting relatively easy grass - it's not like it's getting bogged down and can't catch back up. It just slows down gradually and never picks back up.

Now, I start it up and it runs great. I engage the blades and they come on like a jet engine. It cuts fine for about a lap, and then the blade speed starts dropping off dramatically. It gets to the point where it's spinning but barely cutting anything. I turn the mower off and wait a few minutes, then start it back up and the blade speed is back to normal - until I start cutting, when it does the same thing and slows down after a short while.
I DO hear some squeaking from the middle spindle, which makes me think one of the belts is slipping. I have spares, so I replaced both belts just to eliminate that as a cause, and it didn't make a difference. I tightened the spring on the handle that engages the blades, but I don't see what else I can adjust that might improve the situation.
If it means anything, the engine belt is hot when I touch it, the deck belt is not.
I looked under the deck and there is no grass buildup. The blades slow down even when cutting relatively easy grass - it's not like it's getting bogged down and can't catch back up. It just slows down gradually and never picks back up.

Thoughts?

Ok did you check to see how much movement the deck has does it move forward and backwards

What happen over time the Deck bars pins wear and can cause the belts to slip They do make a bushing kit fix it but those don't last long
Best thing to do is buy the new style bars. They are adjustable

Just checked - the left side is pretty solid, the right side has a little bit of play not a lot, but definitely more than the left. When I start it up and engage the blades, the deck is pretty solid and doesn't really move in any direction.

It had been sitting for close to an hour. When I fired it back up, the blades sounded great, so I decided to mow a little. I got a little more done, and then it started squeaking and the blade speed dropped. It's like the belt is heating up and expanding, causing it to slip. That's the only thing I can think of, and it doesn't make a lot of sense to me.

What's interesting - the chain on the right front leading from the frame to the deck - isn't tight. It's not holding up anything. Left side is tight, right is not. The other chains on the right side are what's holding it up.